Question: Solve for $x$ : $ 4|x - 10| - 8 = 1|x - 10| + 5 $
Solution: Subtract $ {1|x - 10|} $ from both sides: $ \begin{eqnarray} 4|x - 10| - 8 &=& 1|x - 10| + 5 \\ \\ { - 1|x - 10|} && { - 1|x - 10|} \\ \\ 3|x - 10| - 8 &=& 5 \end{eqnarray} $ Add ${8}$ to both sides: $ \begin{eqnarray} 3|x - 10| - 8 &=& 5 \\ \\ { + 8} &=& { + 8} \\ \\ 3|x - 10| &=& 13 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x - 10|} {{3}} = \dfrac{13} {{3}} $ Simplify: $ |x - 10| = \dfrac{13}{3}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -\dfrac{13}{3} $ or $ x - 10 = \dfrac{13}{3} $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -\dfrac{13}{3} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -\dfrac{13}{3} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -\dfrac{13}{3} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $3$ $ x = - \dfrac{13}{3} {+ \dfrac{30}{3}} $ $ x = \dfrac{17}{3} $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = \dfrac{13}{3} $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& \dfrac{13}{3} \\ \\ {+ 10} && {+ 10} \\ \\ x &=& \dfrac{13}{3} + 10 \end{eqnarray} $ Change the ${ + 10}$ to an equivalent fraction with a denominator of $3$ $ x = \dfrac{13}{3} {+ \dfrac{30}{3}} $ $ x = \dfrac{43}{3} $ Thus, the correct answer is $x = \dfrac{17}{3} $ or $x = \dfrac{43}{3} $.